Optimal. Leaf size=50 \[ \frac {1}{72} \sqrt {-x+x^4} \left (-3 x-2 x^4+8 x^7\right )-\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.46, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2046, 2049,
2054, 212} \begin {gather*} -\frac {1}{36} \sqrt {x^4-x} x^4-\frac {1}{24} \sqrt {x^4-x} x+\frac {1}{9} \sqrt {x^4-x} x^7-\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2046
Rule 2049
Rule 2054
Rubi steps
\begin {align*} \int x^6 \sqrt {-x+x^4} \, dx &=\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{6} \int \frac {x^7}{\sqrt {-x+x^4}} \, dx\\ &=-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{8} \int \frac {x^4}{\sqrt {-x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {-x+x^4}-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{16} \int \frac {x}{\sqrt {-x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {-x+x^4}-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{24} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-x+x^4}}\right )\\ &=-\frac {1}{24} x \sqrt {-x+x^4}-\frac {1}{36} x^4 \sqrt {-x+x^4}+\frac {1}{9} x^7 \sqrt {-x+x^4}-\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 66, normalized size = 1.32 \begin {gather*} \frac {\sqrt {x \left (-1+x^3\right )} \left (x^{3/2} \left (-3-2 x^3+8 x^6\right )-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {-1+x^3}}{x^{3/2}}\right )}{\sqrt {-1+x^3}}\right )}{72 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.48, size = 329, normalized size = 6.58 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{72} \, {\left (8 \, x^{7} - 2 \, x^{4} - 3 \, x\right )} \sqrt {x^{4} - x} + \frac {1}{48} \, \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{6} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 56, normalized size = 1.12 \begin {gather*} \frac {1}{72} \, {\left (2 \, {\left (4 \, x^{3} - 1\right )} x^{3} - 3\right )} \sqrt {x^{4} - x} x - \frac {1}{48} \, \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{48} \, \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^6\,\sqrt {x^4-x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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